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Bidisha Roy

Publications and source records attributed to Bidisha Roy.

14 recordsLinked to original sources
arXiv2026

Classification of the rank of a certain family of elliptic curves

In this article, we study the family of elliptic curves $E_{-2pq}: y^2=x^3-2pqx$, where $p$ and $q$ are distinct odd primes. Using a $2$-isogeny methods and some elementary techniques, we obtain explicit possibilities for the Mordell--Weil ranks, conditional on the Parity Conjecture. Moreover, in the rank-one case, we are also able to derive explicit conditions that are independent of the parity conjecture. Moreover, the main results depend only on the residue classes of $(p,q)$ modulo $8$ and the Legendre symbols $\legendre{p}{q}$.

math.NT↗
arXiv2026

On the distribution of shapes of sextic pure number fields

The shape of a number field $K$ of degree $n$ is defined as the equivalence class of the lattice of integers with respect to linear operations that are composites of rotations, reflections, and positive scalar dilations. The shape is a point in the space of shapes $\mathscr{S}_{n-1}$, which is the double quotient $\mathrm{GL}_{n-1}(\mathbb{Z}) \backslash \mathrm{GL}_{n-1}(\mathbb{R}) / \mathrm{GO}_{n-1}(\mathbb{R})$. We investigate the distribution of shapes of pure sextic number fields $K=\mathbb{Q}(\sqrt[6]{m})$, ordered by absolute discriminant. Such fields are partitioned into $20$ distinct Types determined by local conditions at $2$ and $3$, and an explicit integral basis is given in each case. For each Type, the shape of $K$ admits an explicit description in terms of shape parameters. Fixing the sign of $m$ and a Type, we prove that the corresponding shapes are equidistributed along a translated torus orbit in the space of shapes. The limiting distribution is given by an explicit measure expressed as the product of a continuous measure and a discrete measure.

math.NT↗
arXiv2023

Elliptic curve over totally real fields: A Survey

In this survey article, we summarise the known results towards the conjecture: elliptic curves over totally real number fields are modular. For understanding these recent results in the literature, we present some necessary background along with certain applications.

math.NT↗
arXiv2023

Transcendence measure of $e^{1/n}$

For a given transcendental number $\xi$ and for any polynomial $P(X)=: \lambda_0+\cdots+\lambda_k X^k \in \mathbb{Z}[X]$, we know that $ P(\xi) \neq 0.$ Let $k \geq 1$ and $\omega (k, H)$ be the infimum of the numbers $r > 0$ satisfying the estimate $$ \left|\lambda_0+\lambda_1 \xi+\lambda_2 \xi^{2}+ \ldots +\lambda_k\xi^{k}\right| > \frac{1}{H^r}, $$ for all $(\lambda_0, \ldots ,\lambda_k)^T \in \mathbb{Z}^{k+1}\setminus\{\overline{0}\}$ with $\max_{1\le i\le k} \{|\lambda_i|\} \le H$. Any function greater than or equal to $\omega (k, H)$ is a {\it transcendence measure of $\xi$}. In this article, we find out a transcendence measure of $ e^{1/n}$ which improves a result proved by Mahler(\cite{Mahler}) in 1975.

math.NT↗
arXiv2022

Frobenius constants for families of elliptic curves

The paper deals with a class of periods, Frobenius constants, which describe monodromy of Frobenius solutions of differential equations arising in algebraic geometry. We represent Frobenius constants related to families of elliptic curves as iterated integrals of modular forms. Using the theory of periods of modular forms, we then witness some of these constants in terms of zeta values.

math.NT↗
arXiv2022

Plasmonics enabled atomically thin linearly polarized emitter at room temperature

Two-dimensional transition metal di-chalcogenide semiconductors provide unique possibilities to investigate strongly confined excitonic physics and a plasmonic platform integrable to such materials constitutes a hybrid system that can be of interest to enable manipulation of their cumulative optical properties. Here we report tuning of excitonic emission from monolayer WSe2, mechanically exfoliated on top of a periodic two dimensional plasmonic array of elliptical gold (Au) nanodiscs. By exploiting the polarization-dependent nature of plasmonic resonance of the nano plasmonic array (NPA), the photoluminescence (PL) emission from the overlaid monolayer WSe2 could be significantly manipulated. PL is preferentially enhanced at the NPA covered regions of the ake when excited closer to the plasmonic resonant frequencies and previously unpolarized WSe2 PL emission gained ~ 20 up to 40 % degree of linear polarization at room temperature. Obtaining significant spectral overlap between the PL spectrum of WSe2 and the polarization tunable plasmonic resonance of the NPA plays a crucial role in this observation. The results demonstrate active tunability of optical emission from WSe2 by using an otherwise passive plasmonic environment and open the possibility of achieving atomically thin linearly polarized emitters at room temperature. In addition to fundamentally interesting physics of such interactions this can be highly desirable for ultrathin orientation sensitive opto-electronic device related applications.

cond-mat.mes-hall↗
arXiv2022

Moments of Gaussian hypergeometric functions over finite fields

We prove explicit formulas for certain first and second moment sums of families of Gaussian hypergeometric functions $_{n+1}F_n$, $n\ge1$, over finite fields with $q$ elements where $q$ is an odd prime. This enables us to find an estimate for the value $_6F_5(1)$. In addition, we evaluate certain second moments of traces of the family of Clausen elliptic curves in terms of the value $_3F_2(-1)$. These formulas also allow us to express the product of certain $_2F_1$ and $_{n+1}F_n$ functions in terms of finite field Appell series which generalizes current formulas for products of $_2F_1$ functions. We finally give closed form expressions for sums of Gaussian hypergeometric functions defined using different multiplicative characters.

math.NT↗
arXiv2021

Torsion groups of Mordell curves over number fields of higher degree

Mordell curves over a number field $K$ are elliptic curves of the form $ y^2 = x^3 + c$, where $c \in K \setminus \{ 0 \}$. Let $p \geq 5$ be a prime number, $K$ a number field such that $[K:\mathbb{Q}] \in \{ 2p, 3p \}$ and let $E$ be a Mordell curve defined over $K$. We classify all the possible torsion subgroups $E(K)_{\text{tors}}$ for all Mordell curves $E$ defined over $\mathbb{Q}$ when $[K: \mathbb{Q}] \in \{2p, 3p \}$.

math.NT↗
arXiv2019

Torsion groups of Mordell curves over cubic and sextic fields

In this paper, we classify torsion groups of rational Mordell curves explicitly over cubic fields as well as over sextic fields. Also, we classify torsion groups of Mordell curves over cubic fields and for Mordell curves over sextic fields, we produce all possible torsion groups.

math.NT↗
arXiv2018

On Fractionally Dense Sets

In this article, we prove some subsets of the set of natural numbers $\mathbb{N}$ and any non-zero ideals of an order of imaginary quadratic fields are fractionally dense in $\mathbb{R}_{>0}$ and $\mathbb{C}$ respectively.

math.NT↗
arXiv2018

Quadratic non-residues and non-primitive roots satisfying a coprimality condition

Let $q\geq 1$ be any integer and let $ ε\in [\frac{1}{11}, \frac{1}{2})$ be a given real number. In this short note, we prove that for all primes $p$ satisfying $$ p\equiv 1\pmod{q}, \quad \log\log p > \frac{\log 6.83}{\frac{1}{2}-ε} \mbox{ and } \frac{ϕ(p-1)}{p-1} \leq \frac{1}{2} - ε, $$ there exists a quadratic non-residue $g$ which is not a primitive root modulo $p$ such that $gcd\left(g, \frac{p-1}{q}\right) = 1$.

math.NT↗
arXiv2018

On determination of Zero-sum $\ell$-generalized Schur Numbers for some linear equations

Let $r$, $m$ and $k\geq 2$ be positive integers such that $r\mid k$ and let $v \in \left[ 0,\lfloor \frac{k-1}{2r} \rfloor \right]$ be any integer. For any integer $\ell \in [1, k]$ and $ε\in \{0,1\}$, we let $\mathcal{E}_{v}^{(\ell, ε)}$ be the linear homogeneous equation defined by $\mathcal{E}_{v}^{(\ell, ε)}: x_1 + \cdots + x_{k-(rv+ε)} =x_{k-(rv+ε-1)} +\cdots+ \ell x_{k}$. We denote the number $S_{\mathfrak{z},m}^{(\ell, ε)}(k;r;v)$, which is defined to be the least positive integer $t$ such that for any $m$-coloring $χ: [1, t] \to \{0, 1,\ldots,m-1\}$, there exists a solution $(\hat{x}_1, \hat{x}_2, \ldots, \hat{x}_k)$ to the equation $\mathcal{E}_{v}^{(\ell,ε)}$ that satisfies the $r$-zero-sum condition, namely, $\displaystyle\sum_{i=1}^kχ(\hat{x}_i) \equiv 0\pmod{r}$. In this article, we completely determine the constant $S_{\mathfrak{z}, 2}^{(k,1)}(k;r;0)$, $S_{\mathfrak{z}, m}^{(k-1,1)}(k;r;0)$, $S_{\mathfrak{z}, 2}^{(1,1)}(k;2;1)$ and $S_{\mathfrak{z}, r}^{(1,0)}(k;r;v)$. Also, we prove upper bound for the constants $S_{\mathfrak{z},2}^{(2,1)}(k;2;0)$ and $S_{\mathfrak{z},2}^{(1,1)}(k;2;v)$.

math.CO↗
arXiv2018

The Determination of 2-color zero-sum generalized Schur Numbers

Consider the equation $\mathcal{E}: x_1+ \cdots+x_{k-1} =x_{k}$ and let $k$ and $r$ be positive integers such that $r\mid k$. The number $S_{\mathfrak{z},2}(k;r)$ is defined to be the least positive integer $t$ such that for any 2-coloring $χ: [1, t] \to \{0, 1\}$ there exists a solution $(\hat{x}_1, \hat{x}_2, \ldots, \hat{x}_k)$ to the equation $\mathcal{E}$ satisfying $\displaystyle \sum_{i=1}^kχ(\hat{x}_i) \equiv 0\pmod{r}$. In a recent paper, the first author posed the question of determining the exact value of $S_{\mathfrak{z}, 2}(k;4)$. In this article, we solve this problem and show, more generally, that $S_{\mathfrak{z}, 2}(k, r)=kr - 2r+1$ for all positive integers $k$ and $r$ with $k>r$ and $r \mid k$.

math.CO↗